Does an electron undergo a form of Brownian motion in the vacuum? There are some questions on StackExchange such as this one
Brownian Motion in Vacuum
asking about Brownian motion in the vacuum. There are related papers such as this one:
https://arxiv.org/abs/quant-ph/9808032 (Found.Phys. 29 (1999) 1917-1949)
The idea is that we have a charged particle or a mirror with, around it, the electromagnetic field in its ground state. The claim is that the particle or mirror will undergo a combination of diffusive and viscous motion, like Brownian motion, owing to its interaction with the field. I doubt this. But I note that a little cottage industry of scientific work has popped up, in which it is claimed that such motion is the prediction of standard physics. I doubt it because I can't see how it would conserve energy. The claim is not that the electron or mirror heats indefinitely, but that it reaches a finite 'temperature'. However if this really corresponded to a fluctuating motion then I think such an electron or mirror would radiate photons (real ones!) However the calculations in a paper such as the one mentioned above seem thorough and careful, so what is going on?
It could be that the authors have chosen for the initial state of motion of the charged particle or mirror a state which has the appearance of being an inertial state of motion, but is not an eigenstate of the Hamiltonian of all the relevant fields together. In that case one would expect evolution over time. But I think that if $|\Omega \rangle$ is the ground state of all the interacting fields (electromagnetic, Dirac, etc.) then there is a state
$$
\hat{A}^\dagger | \Omega \rangle  \tag{1}
$$
which is an eigenstate of the total Hamiltonian and which has one electron present. The operator $\hat{A}^\dagger$ here is a kind of raising operator, but it does not introduce just an excitation of the bare Dirac field (without interactions). It introduces an excitation of all the fields together in such a way that in the result there is one electron present. Such a state would not, I think, undergo Brownian motion.
To be specific, then:

*

*Is the state labeled (1) above a valid use of the concepts, and is it indeed a state which is an eigenstate of the total Hamiltonian, in which one electron is present?


*Am I correct that such a state does not correspond to anything which could be called Brownian motion (it would of course have non-zero $\Delta x \Delta p$ but that is not the issue here).


*Are there other states which could be correctly described as an electron (or a mirror) in otherwise empty space undergoing Brownian motion into the infinite future?
I have general expertise in physics and some general knowledge of quantum field theory; I guess it would need someone with a greater level of expertise in quantum field theory to provide an answer to this question.
 A: 
it is claimed that such motion is the prediction of standard physics. I doubt it because I can't see how it would conserve energy. The claim is not that the electron or mirror heats indefinitely, but that it reaches a finite 'temperature'. However if this really corresponded to a fluctuating motion then I think such an electron or mirror would radiate photons (real ones!)

I think the idea is that a localized electron or "mirror" in vacuum (no other close sources of EM fields) would manifest brownian motion (random changes of velocity, or in QT, expectation value of velocity). I think it is quite plausible. Both in classical and quantum theories, EM field plausibly has non-zero variance even in the best of vacuums (background radiation, uncertainty relations...) Only such motion in such hypothetical scenario is hard to observe, because any instrument will introduce its own fields.
Modelling this using a Hamiltonian and its eigenstate does not seem meaningful; such eigenstate evolves only phase, in linear manner, which is not chaotic in any way.
I think your expectation about the charge itself radiating due to such jittery motion is correct; from Maxwell's equations and also experience we know that acceleration of the whole localized charged particle in single direction means radiation is produced.
I don't see a problem with energy conservation in classical class of theories; it is sufficient to give up on Poynting expressions, which were derived for regular finite charge distributions, but they fail spectacularly for singular charges like point or line charge. There is a consistent theory of point particles due to Frenkel where EM energy density is a sum of bilinear terms in all particle field pairs, and this density can be positive or negative depending on orientation of all the particle fields.
If we model the process quantum-theoretically as non-eigenstate being evolved by some Hamiltonian of the whole system of all fields, then we don't have energy value defined in general, only its expectation value with some variance, and these are constant in time, while expectation of electron's velocity need not be.
If we model this classically, zero point background EM radiation can provide or extract all EM energy needed to explain changes in electron's motion due to fluctuating EM field. In other words, the quantum vacuum (or background EM radiation) can supply or extract energy to maintain energy conservation.
